# Simplifying Expressions – GCSE Maths Insight of the Week 1

Each week we set all our Year 11 Higher GCSE students a GCSE Essential Skills Quiz from my Diagnostic Questions website. These quizzes are completely free and are designed to test the content that appears on the majority of GCSE higher maths papers. These are the kind of questions that students should get right, but which time and time again cost our Year 11s the valuable marks that prevent them getting the grade they want.

We set our quizzes on a Monday, the students complete them for Friday, and then I sit down with a Mellow Birds coffee and a yoghurt (it is the weekend, after all) to analyse the results. I try to find one key insight, and then share it with staff in our weekly Departmental Meeting in a 5 minute slot called “Insight of the Week”.

Why not join in too by setting these quizzes to your Year 11s? The more the merrier!

Quiz 1 is below, followed by the key insight from this week.

How our students performed compared to the rest of the world:

The Insight of the Week comes from the question that caused our students the most problems, which was on simplifying algebraic expressions:

Only 37% of our students got this question correct, with C being the most common answer chosen, luring in 47% of them.

Let’s look at some of our students’ explanations to see if we can understand why:

Incorrect Answer A (10% of our students chose this)

This may be students mistaking the rules for multiplying with the rules for addition. Many student explanations for this choice of answer also highlight their compulsion to combine the powers of d together.

2×3=6 DxD=D squared D squared add D = D cubed

Because d is 1 so if there’s 2 ds and one spare then 3ds

Incorrect Answer B (6% of our students chose this)

Students opting for this answer believe you can add the numbers together and add the powers together, but interestingly also may have some notion of the need to keep numbers and letters separate

3+2=5 (d squared)

d2 and d =d3

Incorrect Answer C (47% of our students chose this)

By far the most popular choice. Students are determined to combine the terms in a way that seems the most sensible: add the numbers and add the powers.

i think this because 2 and 3 is 5 and if you add the d’s on you get in a total of 3 d’s

3 plus 2 is 5 and d to the power of 2 plus another d is d to the power of 3

3 + 2 = 5 d squared + d = d cubed d cubed + 5 = 5d3 (d cubed)

One of my motivations for developing Diagnostic Questions was so that students all around the world cold learn from each other. So, what are our Year 11 students’ favourite correct explanations to help them resolve their own misconceptions?

because d squared and d are different things so you can’t put them together because it wouldn’t make sense

you cannot group these terms together – d and d squared are two completely different things.”

because you can only collect like terms and these aren’t like terms

Tackling the Misconception in Class

So, when we discussed this insight in our Departmental Meeting, what ideas did we come up with for resolving the misconceptions in lessons this coming week with our students:

• Substitute numbers in for d to show that the expressions are not equal
• Explicitly name the terms different things – “d” and “d squared”
• Write the expressions out in full, so 2d^2 would be 2 x d x d
• Write the expressions in the form d^2 + d^2 and d + d + d, and compare this to a + a and b + b + b

Please add any extra ideas in the Comments section below!

The series of GCSE Essential Skills Quizzes are available hereand will always be completely free.

Quiz 2 is available here

I hope this quizzes will prove useful to help your students develop the essential skills necessary for success at GCSE, and aid your teachers gain a deeper understanding of how your students learn.

# Year 7 Baseline Maths Tests – Assessing without levels

Recently, I have been doing a lot of thinking about assessment – as you can imagine, my wife has been delighted. In my series of blog posts about writing our new maths scheme of work, I talked about our department’s plans for assessing each year group throughout the year. But a more pressing issue is how to assess our lovely new Year 7s.

In the past we have used a variety of commercial packages that return a “maths age” – but I have never been all that impressed by that. For me, the concept of a maths age is as redundant and meaningless as the concept of a level. So, one child has a maths age of 11.4, and another has a maths age of 11.8. What does that tell me about their understanding of place value, or their range of strategies for multiplying two digit numbers together? Well, it tells me about as much as knowing one student is a Level 4a and the other is a 4b – i.e., nothing at all.

Inspiring Blog Posts

Fortunately, I have been inspired by two of the best blog posts I have read all year. Firstly, William Emney wrote a stunning piece entitled “The problem with levels- gaps in basic numeracy skills identified by rigorous diagnostic testing“, about his strategy for diagnostically testing each of his school’s Year 7 students and the amazing insights that he gained from this. Secondly, there was a wonderful summary of the leaked Commission on Assessment without Levels report, by Harry Fletcher-Wood. In his post, Harry beautifully sums up the key points of the report, and crucially the disastrous affect levels have had on schools, teachers and students.

Free Year 7 Maths Baseline Tests

So, I decided to try something different. I want immediate results, but also the opportunity to gain deep insights across the whole year group, highlighting areas of strength and weakness both in individual students and given cohorts. I want to identify interesting and important patterns and trends in specific areas of maths. I want this presented beautifully, so I can get an instantaneous picture, and then delve deeper for greater understanding. And I want all of this without having to do mountains of marking. If only there were a website that could do that?…

So, we are going to use the following baseline assessments with all our Year 7 students. I thought I would share them, in case they are useful to other schools. Oh, and it is all completely free!

I have created a series of ten, 20-question Year 7 baseline diagnostic tests using questions from my Diagnostic Questions website. You can use as many or as few of the tests as you like. The 200 questions are all non-calculator and are designed to be all a similar level of challenge and curriculum coverage. They are mainly number-based, covering the full range of numeracy that the students will have encountered at primary school, with some basic properties of shape, co-ordinates and simple algebraic substitution questions in there as well.

Here is the second one so you can get an idea of the content:

What do the results look like?

Now, these tests will not give you a maths age or a level. However, as well as giving you class and individual student scores and identifying specific questions that caused difficulty, they will also give you something like this:

This a visual representation of a selection of 10 and 11 year old’s understanding of mathematics. The inner rings contain the four key areas (Number, Algebra, Shape and Statistics), and the outer areas break these subjects down, firstly into things like Basic Arithmetic, and then even further into Place Value, Order of Operations, Multiplication, etc. The more green, the better the level of understanding the more red, the bigger the problem. You can have a play around with this data set by clicking this link.

Imagine this was the data for all the Year 7s in your school at the start of the new term. What would it tell you?

Well, we could take a closer look at number:

That will show us, very clearly, that this group of students has some serious problems with BODMAS:

And if we now take a look at the bottom of the page, we can see exactly which questions are causing the problems, and which students are having difficulty:

We can then continue, looking at other areas of weakness, identifying key trends and patterns.

We also have the option to compare two particular classes’ pictures of understanding:

Or an individual student relative to another student or the rest of the class:

Then how about Year 7 girls against Year 7 boys:

Or, how about generating one of these diagrams for each of your students?

What to do with the results?

Now, obviously I am completely biased, but I think this has potential to generate some seriously rich, meaningful data that actually tells you something. Things like:

• It may identify key areas that you as a department want to tackle in those first few weeks in September to give your students a solid mathematical foundation for the year
• It may highlight interesting features about specific topics, for example which methods of written multiplication students are most comfortable with from primary school
• It may reveal key gaps in knowledge, but also topics that the whole year group is entirely comfortable with, allowing you to allocate your teaching time as efficiently and productively as possible.
• It may help to identify key groups of students who require specific intervention
• When your students review their results, reading and liking other students’ explanations, it will give you insights into how students understand the topics via the most popular explanations chosen, whilst at the same time helping them to resolve their own misconceptions.
• And then, a few weeks later, perhaps you could test the students again, either using the same 10 tests, or any left over if you did not use them all. This will give you a really clear measure of the progress the students have mad,e and what further interventions may be needed.

How to run the tests?

Of course, expecting Year 7 students to answer 200 questions all in one go, giving full explanations for each, is asking a bit much, and might hinder their performance and the insights you gain from it. So, after much consideration, we are going to run the baseline tests over two maths lessons, using computer rooms, encouraging the students to only give explanations for the questions that they are not sure about.

Other suggestions for using the baseline tests might be:

• Only use 5 of the quizzes (possibly use the other 5 at a later date as a test of progress)
• Tell students they don’t need to give any explanations
• Tell students to explain just two questions every quiz
• Run it as a “live quiz” in the classroom – or with all Year 7s together in the Hall if you are feeling really brave!
• Set some of the quizzes as a homework – although, of course, you cannot be sure it is all their own work

Accessing the Maths Baseline Tests

For instructions on how to add new students to Diagnostic Questions, or to move students between classes, please see this post.

Here are the links to all 10 Year 7 Baseline Assessments so you can assign them to your students:

I really hope you, your department and your students find them useful. Please let me know how you get on in the Comments below.

And just a reminder, I am running a new “Assessment for Learning with Diagnostic Questions” course from September. Please visit this page for more details

# My new AfL with Diagnostic Questions workshop

From September 2015, on Wednesday afternoons and all day Thursday, I will be available to deliver a brand-new, Assessment for Learning with Diagnostic Questionsworkshop in your school. This will be a high-quality (I promise!) CPD event that will be jam-packed with effective, simple classroom-ready strategies, working together with the unique power of Diagnostic Questions, to help you identify, understand and resolve your students’ key misconceptions.

Any schools who book a workshop will also receive a free one-year subscription to our exciting, new Premium Features.

And, as an extra incentive, there will be special deals on the workshop for any September bookings.

# Diagnostic Question Badges and Reputation Points

Over at Diagnostic Questions, we have introduced a new feature, whereby students and teachers (because I know you wouldn’t want to be left out!), can earn badges and reputation points based on their levels of contributions to the Diagnostic Questions community. Here are some of the many highly-exclusive badges up for grabs:

And I am blaming Simon entirely for the “Doffing Hat” – I am already getting abuse from my Year 8s about that one.

As well as badges, we also have reputation points. How are reputation points calculated, I hear you ask? Well, much like the recipe for Kentucky Fried Chicken, it remains a closely guarded secret! But the formula is designed to reward valuable contributions to the Diagnostic Questions community, so things like writing good explanations, creating questions and quizzes, regularly visiting the site to answer questions and helping us spread the word about Diagnostic Questions will all see your points begin to accumulate.

If you haven’t visited the site for a while, please go and have a look around. We have made loads of improvements, not least of which involves adding all the other school subjects, so please tell your colleagues. And if you want a sneak preview of 3 premium features coming to the site next year, then have a read of this post.

Thanks for all your help and support with the website.

Craig

# Diagnostic Questions: exciting news

It gives me great pleasure to unveil a sneak preview of three brand-new, exciting features that are coming toDiagnostic Questions in the new academic year.

As we have said from the outset, we want to keep the core functionality of Diagnostic Questions free to all users. That means it will remain free for teachers to access thousands of questions, create your own questions, build quizzes, assign these quizzes to your classes, view your students’ answers and explanations, and compare these results to other classes. Likewise, it will remain free for students to use the website to revise independently, answering thousands of questions and benefiting from reading explanations written by students all around the world until they find one that makes sense to them.

We have already seen how these free features have helped teachers and students from all over the world identify, understand and resolve key misconceptions. We currently have over 6,000 free maths questions, and close to 1 million answers, giving us a unique insight into how students learning. We now cover all other subjects as well, so please tell your colleagues about us!

To build this into a sustainable business, we are developing three brand new Premium features that we believe will make Diagnostic Questions even more powerful:

1. Teacher-Student Dialogue and Feedback

We are acutely aware of the Ofsted requirement for personalised, effective dialogue and feedback between student and teacher. This is often lacking in online products. So, we are building a system whereby when a student answers a question and gives their explanation, you as the teacher are able to respond with personalised feedback and then set a follow-up question to check the misconception has been resolved. This will form a digital record of progress which can be printed off at any time.

2. Data Insights

The more your students answer questions and give explanations, the more the website can tell you about their understanding of mathematics. Whilst you can already learn about your classes’ understanding of a given question or quiz, we can do so much more. So, at the touch of a button we will be able to tell you things like:
• What are the 5 more important areas for improvement for your class?
• Which students should you pair together to get effective peer-to-peer learning?
• What specific areas should each student focus on?
• What evidence is there of the progress your class has made over a set period of time?

We believe that these insights, and more that will follow, will help make teaching and learning more effective than ever.

3. Reporting

At the click of a button you will be able to generate a detailed, personalised, insightful report on your class, an individual student, or a given cohort of students. So, imagine you want to know how Year 7 girls are performing in Algebra, compared to Year 7 boys, then we can do that. Or you want a detailed, personalised summary, including key action points, for each student in your Year 10 class, then that is no problem as well.

We have a special, early-bird price for these features of £500 for your entire school, covering all subjects, for the 2015/16 academic year.

We hope that these features will make Diagnostic Questions, and the insights you can gain from the website, even more powerful, and this have a greater positive effect on assessment, teaching and learning.

If you want to know more, or have any questions, or want to place an order, just drop us Simon and I an email via this link